ch3l4 survival analysis
-
Upload
monandiogega -
Category
Documents
-
view
30 -
download
3
description
Transcript of ch3l4 survival analysis
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Cox proportional hazards modelST3242: Introduction to Survival Analysis
Alex Cook
September 2008
ST3242 : Cox proportional hazards model 1/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
NUS news
ST3242 : Cox proportional hazards model 2/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
NUS news
ST3242 : Cox proportional hazards model 3/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
NUS news
ST3242 : Cox proportional hazards model 4/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
NUS news
ST3242 : Cox proportional hazards model 5/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
NUS news
ST3242 : Cox proportional hazards model 6/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
NUS news
ST3242 : Cox proportional hazards model 7/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Plan for today
Go over mock mid-term test
3pm feedback
Lecture: confidence intervals for the Cox phm
ST3242 : Cox proportional hazards model 8/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Exam advice
Read all questions at start of exam
Read all instructions
Ensure do everything asked and nothing more
Write as neatly as you can
Don’t “do a Ravi”
Avoid rounding too much too soon
ST3242 : Cox proportional hazards model 9/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Exam advice
INSTRUCTIONS TO CANDIDATES:
1 This test contains EIGHT questions and comprisesFOUR printed pages.
2 Answer ALL questions for a total of 100 marks.
3 This is a closed-book test; only non-programmablecalculators are allowed.
ST3242 : Cox proportional hazards model 10/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q1: [3]
If the survival function is S(t), what is the hazard functionh(t)?
ST3242 : Cox proportional hazards model 11/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q2: [6]
If the hazard function is h(t) = θt for a parameter θ > 0,what are the survival and density functions?
ST3242 : Cox proportional hazards model 12/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q3: [10]
If survival times in the absence of censoring are distributedaccording to a log-logistic distribution with parameters κ andλ, the hazard and survival functions are
h(t) =λκtκ−1
1 + λtκ
S(t) =1
1 + λtκ
respectively. If we have observed data of the form (ti , δi),where δi = 1 if individual i fails at time ti and δi = 0 if i isright-censored at ti , for i = 1, . . . ,m, what is thelog-likelihood function?
ST3242 : Cox proportional hazards model 13/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q4: [6]
Consider a study into the length of time people can survivewithout purchasing or receiving plastic goods. The studystarts at time Tstart and ends at time Tend. Three nusstudents are among the participants.
Student A is recruited to the study at time Tstart. Shebuys no plastic until time tA < Tend, when she buys abottle of Pocari Sweat.
For each student, say if the datum is uncensored,right-censored, left-censored or interval-censored.
ST3242 : Cox proportional hazards model 14/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q4: [6]
Consider a study into the length of time people can survivewithout purchasing or receiving plastic goods. The studystarts at time Tstart and ends at time Tend. Three nusstudents are among the participants.
Student B is recruited to the study at time Tstart. Hereports to the organisers at time tB < Tend that he hasbought no plastics, but they never hear from him again.
For each student, say if the datum is uncensored,right-censored, left-censored or interval-censored.
ST3242 : Cox proportional hazards model 15/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q4: [6]
Consider a study into the length of time people can survivewithout purchasing or receiving plastic goods. The studystarts at time Tstart and ends at time Tend. Three nusstudents are among the participants.
Student C is recruited to the study at timetC ∈ (Tstart,Tend). She buys no plastic for the remainderof the study.
For each student, say if the datum is uncensored,right-censored, left-censored or interval-censored.
ST3242 : Cox proportional hazards model 16/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q5: [12]
Use the delta method to approximate the mean and varianceof√
X , where X is a random variable with mean µ andvariance σ2. When do you expect these will provide a goodapproximation to the true mean and variance of
√X?
ST3242 : Cox proportional hazards model 17/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q6(i): [36]
Calculate and sketch the Kaplan–Meier estimate of S(t) forthe following data, where δi = 1 if individual i died at time tiand δi = 0 if i was censored at that time.
i ti δi1 0.6 12 0.9 13 1.1 04 1.5 15 2.0 06 2.0 0
ST3242 : Cox proportional hazards model 18/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q6(ii): [36]
Recall that
V{S(t)} = S(t)2∑t(i)≤t
d(i)
n(i−)(n(i−) − d(i))
where t(i) is the ith ordered event time, d(i) is the number ofdeaths at that time and n(i−) is the number at risk an instantbefore that time.Construct the “naıve” 95% confidence interval described inthe lecture notes for S(1), the probability of surviving at leastto one time unit. What is the probability the true value ofS(1) lies within your confidence interval?
ST3242 : Cox proportional hazards model 19/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q7: [12]
The following are data on the times to recurrence of breastcancer following treatment in German women. These datahave been grouped according to whether the patients haveundergone the menopause (g = 2) or not (g = 1); t(i) is thetime in years until the ith unique recurrence time, ng ,(i−) is thetotal number at risk in group g an instant before time t(i),dg ,(i) is the number in group g that had a recurrence at timet(i), while n(i−) = n1,(i−) + n2,(i−) and d(i) = d1,(i) + d2,(i). Thecolumns marked eg ,(i) and vg ,(i) are the expected value andvariance of dg ,(i) under the hypothesis that menopause isindependent of recurrence, respectively. Note that for yourconvenience, the data have been grouped into years. [To findout more about these data, see Hosmer et al., 2008.]
ST3242 : Cox proportional hazards model 20/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q7: [12]
i t(i) n1,(i−) n2,(i−) n(i−) d1,(i) d2,(i) d(i) e1,(i) e2,(i) v1,(i) v2,(i)
1 1 290 396 686 29 27 56 23.7 32.3 12.6 12.62 2 245 357 602 44 65 109 44.4 64.6 21.6 21.63 3 183 275 458 20 39 59 23.6 35.4 12.4 12.44 4 140 191 331 16 23 39 16.5 22.5 8.4 8.45 5 98 130 228 4 18 22 9.5 12.5 4.9 4.96 6 56 65 121 6 5 11 5.1 5.9 2.5 2.57 7 14 22 36 0 3 3 1.2 1.8 0.7 0.7
Test the hypothesis that the menopause is independent ofbreast cancer recurrence.
ST3242 : Cox proportional hazards model 21/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q8(i): [15]
Given that the Cox proportional hazards model for a singlecontinuous covariate xi is
h(t, xi) = h0(t,α) exp{βxi}
for individual i , what is the corresponding survival functionincorporating this covariate?
ST3242 : Cox proportional hazards model 22/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Q8(ii): [15]
What is the hazard ratio comparing an individual withcovariate x1 to one with covariate x2 at time t?
ST3242 : Cox proportional hazards model 23/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β, hazard ratios
So far. . .
Saw an equation for asymptotic distribution of β:
β ∼ N(β,E{I(β)}−1)
Can replace by observed information:
β ∼ N(β, I(β)−1)
R gives us standard errors for the individual βs
ST3242 : Cox proportional hazards model 24/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β, hazard ratios
So far. . .
Saw an equation for asymptotic distribution of β:
β ∼ N(β,E{I(β)}−1)
Can replace by observed information:
β ∼ N(β, I(β)−1)
R gives us standard errors for the individual βs
ST3242 : Cox proportional hazards model 24/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β, hazard ratios
So far. . .
Saw an equation for asymptotic distribution of β:
β ∼ N(β,E{I(β)}−1)
Can replace by observed information:
β ∼ N(β, I(β)−1)
R gives us standard errors for the individual βs
ST3242 : Cox proportional hazards model 24/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β, hazard ratios
ST3242 : Cox proportional hazards model 25/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β, hazard ratios
We want more!
CIs for β
CIs for hazard ratio between two categories
CIS for hazard ratio between two individuals
ST3242 : Cox proportional hazards model 26/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β
The asymptotic distribution of β can be estimated using theobserved information
β ∼ N(β, I(β)−1)
If β is vector of length p, I(β) is p × p matrixi , j element is
− d2lp(β)
dβi dβj
ST3242 : Cox proportional hazards model 27/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
β1
β2
β3
∼ N
β1
β2
β3
,
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
where
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
=
− d2lp(β)
dβ1 dβ1− d2lp(β)
dβ1 dβ2− d2lp(β)
dβ1 dβ3
− d2lp(β)dβ2 dβ1
− d2lp(β)dβ2 dβ2
− d2lp(β)dβ2 dβ3
− d2lp(β)dβ3 dβ1
− d2lp(β)dβ3 dβ2
− d2lp(β)dβ3 dβ3
−1
ST3242 : Cox proportional hazards model 28/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
If β1
β2
β3
∼ N
β1
β2
β3
,
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
the marginal distribution of β1 is
β1 ∼ N(β1, σ11)
R
gives√σii as standard output!
ST3242 : Cox proportional hazards model 29/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
If β1
β2
β3
∼ N
β1
β2
β3
,
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
the marginal distribution of β1 is
β1 ∼ N(β1, σ11)
R
gives√σii as standard output!
ST3242 : Cox proportional hazards model 29/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Confidence interval for β
p(β1 − 1.96
√σ11 < β1 < β1 + 1.96
√σ11
)= 0.95
ST3242 : Cox proportional hazards model 30/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Confidence interval for β
p(β1 − 1.96
√σ11 < β1 < β1 + 1.96
√σ11
)= 0.95
ST3242 : Cox proportional hazards model 31/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
A 95% confidence interval for βbmi is
(−0.0985− 1.96× 0.0148,−0.0985 + 1.96× 0.0148)
= (−0.13,−0.07)
ST3242 : Cox proportional hazards model 32/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Confidence interval for g(β)
If (a, b) is a confidence interval for β,
(ea, eb) is a confidence interval for eβ
(eax , ebx) is a confidence interval for eβx
What about a CI for eβ1−β2?
ST3242 : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Confidence interval for g(β)
If (a, b) is a confidence interval for β,
(ea, eb) is a confidence interval for eβ
(eax , ebx) is a confidence interval for eβx
What about a CI for eβ1−β2?
ST3242 : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Confidence interval for g(β)
If (a, b) is a confidence interval for β,
(ea, eb) is a confidence interval for eβ
(eax , ebx) is a confidence interval for eβx
What about a CI for eβ1−β2?
ST3242 : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Confidence interval for g(β)
If (a, b) is a confidence interval for β,
(ea, eb) is a confidence interval for eβ
(eax , ebx) is a confidence interval for eβx
What about a CI for eβ1−β2?
ST3242 : Cox proportional hazards model 33/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
h(t, xla, xad, xsm, xsq) = h0(t) exp(βlaxla+βadxad+βsmxsm+βsqxsq)
where xc = 1 if the individual has cell type c and xc = 0 ifnot. Output relative to one category (adeno) as baseline
ST3242 : Cox proportional hazards model 34/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
h(t, xla, xad, xsm, xsq) = h0(t) exp(βlaxla+βadxad+βsmxsm+βsqxsq)
where xc = 1 if the individual has cell type c and xc = 0 ifnot. Output relative to one category (adeno) as baseline
ST3242 : Cox proportional hazards model 34/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Multivariate normal distribution
If x ∼ N(µ,Σ) then any subset of the rows of x also has aNormal distribution with corresponding rows of µ and rowsand columns of Σ.
For example
β1
β2
β3
∼ N
β1
β2
β3
,
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
⇒(β1
β3
)∼ N
((β1
β3
),
(σ11 σ13
σ31 σ33
))
ST3242 : Cox proportional hazards model 35/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Multivariate normal distribution
If x ∼ N(µ,Σ) then any subset of the rows of x also has aNormal distribution with corresponding rows of µ and rowsand columns of Σ.
For example
β1
β2
β3
∼ N
β1
β2
β3
,
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
⇒(β1
β3
)∼ N
((β1
β3
),
(σ11 σ13
σ31 σ33
))
ST3242 : Cox proportional hazards model 35/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Q: what is V(X + Y )?
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X ,Y )
Q: what is V(X − Y )?
ST3242 : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Q: what is V(X + Y )?
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X ,Y )
Q: what is V(X − Y )?
ST3242 : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Q: what is V(X + Y )?
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X ,Y )
Q: what is V(X − Y )?
ST3242 : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Q: what is V(X + Y )?
A: V(X + Y ) = V(X ) + V(Y ) if X & Y independent
A: V(X + Y ) = V(X ) + V(Y ) + 2C(X ,Y )
Q: what is V(X − Y )?
ST3242 : Cox proportional hazards model 36/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Properties of the covariance
C(X ,Y ) = C(Y ,X )
C(X ,X ) = V(X )
C(aX + b, cY + d) = acC(X ,Y )
Q: what is V(X − Y )?
A:
V(X − Y ) = V(X ) + V(−Y ) + 2C(X ,−Y )
= V(X ) + V(Y )− 2C(X ,Y )
ST3242 : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Properties of the covariance
C(X ,Y ) = C(Y ,X )
C(X ,X ) = V(X )
C(aX + b, cY + d) = acC(X ,Y )
Q: what is V(X − Y )?
A:
V(X − Y ) = V(X ) + V(−Y ) + 2C(X ,−Y )
= V(X ) + V(Y )− 2C(X ,Y )
ST3242 : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Properties of the covariance
C(X ,Y ) = C(Y ,X )
C(X ,X ) = V(X )
C(aX + b, cY + d) = acC(X ,Y )
Q: what is V(X − Y )?
A:
V(X − Y ) = V(X ) + V(−Y ) + 2C(X ,−Y )
= V(X ) + V(Y )− 2C(X ,Y )
ST3242 : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Properties of the covariance
C(X ,Y ) = C(Y ,X )
C(X ,X ) = V(X )
C(aX + b, cY + d) = acC(X ,Y )
Q: what is V(X − Y )?
A:
V(X − Y ) = V(X ) + V(−Y ) + 2C(X ,−Y )
= V(X ) + V(Y )− 2C(X ,Y )
ST3242 : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Properties of the covariance
C(X ,Y ) = C(Y ,X )
C(X ,X ) = V(X )
C(aX + b, cY + d) = acC(X ,Y )
Q: what is V(X − Y )?
A:
V(X − Y ) = V(X ) + V(−Y ) + 2C(X ,−Y )
= V(X ) + V(Y )− 2C(X ,Y )
ST3242 : Cox proportional hazards model 37/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Properties of Cov
Properties of the covariance
C(X ,Y ) = C(Y ,X )
C(X ,X ) = V(X )
C(aX + b, cY + d) = acC(X ,Y )
Q: what is V(βi − βj)?
A:
V(βi − βj) = V(βi) + V(−βj) + 2C(βi ,−βj)
= V(βi) + V(βj)− 2C(βi , βj)
ST3242 : Cox proportional hazards model 38/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for βi − βj
A 95% CI for βi − βj is
βi − βj ± 1.96√σii + σjj − 2σij
R does it for us!
The coxph functions silently outputs var, thevariance–covariance matrixphm.cell=coxph(Surv(t,delta)∼factor(Cell))phm.cell$var
phm.cell$coefficients
ST3242 : Cox proportional hazards model 39/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for βi − βj
A 95% CI for βi − βj is
βi − βj ± 1.96√σii + σjj − 2σij
R does it for us!
The coxph functions silently outputs var, thevariance–covariance matrixphm.cell=coxph(Surv(t,delta)∼factor(Cell))phm.cell$var
phm.cell$coefficients
ST3242 : Cox proportional hazards model 39/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
ST3242 : Cox proportional hazards model 40/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
95%CI for βsq − βla is
−0.77− 1.00± 1.96√
0.0642 + 0.0638− 2× 0.0256
i.e. (−0.31, 0.77)A 95% CI for the hazard ratio comparing squamous to largecells is
(e−0.31, e0.77) = (0.73, 2.17)
ST3242 : Cox proportional hazards model 41/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for β, hazard ratios
We want more!
CIs for β
CIs for hazard ratio between two categories
CIS for hazard ratio between two individuals
ST3242 : Cox proportional hazards model 42/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for hazard ratiosSuppose we want a hazard ratio comparing individuals with xa
and xb
squamous cell, performance of 60, aged 70, with no priortherapy
large cell, performance of 20, aged 50, with no priortherapy
Same approach
obtain β and covariance matrix via coxph
evaluate βTxa and β
Txb
evaluate V(βTxa − β
Txb)
95% CI is βTxa − β
Txb ±
√V(β
Txa − β
Txb)
ST3242 : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for hazard ratiosSuppose we want a hazard ratio comparing individuals with xa
and xb
squamous cell, performance of 60, aged 70, with no priortherapy
large cell, performance of 20, aged 50, with no priortherapy
Same approach
obtain β and covariance matrix via coxph
evaluate βTxa and β
Txb
evaluate V(βTxa − β
Txb)
95% CI is βTxa − β
Txb ±
√V(β
Txa − β
Txb)
ST3242 : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for hazard ratiosSuppose we want a hazard ratio comparing individuals with xa
and xb
squamous cell, performance of 60, aged 70, with no priortherapy
large cell, performance of 20, aged 50, with no priortherapy
Same approach
obtain β and covariance matrix via coxph
evaluate βTxa and β
Txb
evaluate V(βTxa − β
Txb)
95% CI is βTxa − β
Txb ±
√V(β
Txa − β
Txb)
ST3242 : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for hazard ratiosSuppose we want a hazard ratio comparing individuals with xa
and xb
squamous cell, performance of 60, aged 70, with no priortherapy
large cell, performance of 20, aged 50, with no priortherapy
Same approach
obtain β and covariance matrix via coxph
evaluate βTxa and β
Txb
evaluate V(βTxa − β
Txb)
95% CI is βTxa − β
Txb ±
√V(β
Txa − β
Txb)
ST3242 : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for hazard ratiosSuppose we want a hazard ratio comparing individuals with xa
and xb
squamous cell, performance of 60, aged 70, with no priortherapy
large cell, performance of 20, aged 50, with no priortherapy
Same approach
obtain β and covariance matrix via coxph
evaluate βTxa and β
Txb
evaluate V(βTxa − β
Txb)
95% CI is βTxa − β
Txb ±
√V(β
Txa − β
Txb)
ST3242 : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
CIs for hazard ratiosSuppose we want a hazard ratio comparing individuals with xa
and xb
squamous cell, performance of 60, aged 70, with no priortherapy
large cell, performance of 20, aged 50, with no priortherapy
Same approach
obtain β and covariance matrix via coxph
evaluate βTxa and β
Txb
evaluate V(βTxa − β
Txb)
95% CI is βTxa − β
Txb ±
√V(β
Txa − β
Txb)
ST3242 : Cox proportional hazards model 43/44
Soya beans Test Confidence intervals CI for β MVNs, Var, Cov
Example
xa = (xa1 , x
a2 ) and xb = (xb
1 , xb2 ) Variance of βxa − βxb is
V(βxa − βxb) = V{β1(xa1 − xb
1 ) + β2(xa2 − xb
2 )}= (xa
1 − xb1 )2V(β1) + (xa
2 − xb2 )2V(β2)
+2C{β1(xa1 − xb
1 ), β2(xa2 − xb
2 )}= (xa
1 − xb1 )2V(β1) + (xa
2 − xb2 )2V(β2)
+2(xa1 − xb
1 )(xa2 − xb
2 )C{β1, β2}
ST3242 : Cox proportional hazards model 44/44